NAG Library Function Document

nag_matop_complex_gen_matrix_cond_sqrt (f01kdc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_matop_complex_gen_matrix_cond_sqrt (f01kdc) computes an estimate of the relative condition number,

κ_{A^{1 / 2}}

, and a bound on the relative residual, in the Frobenius norm, for the square root of a complex

n

n

matrix

A

. The principal square root,

A^{1 / 2}

, of

A

is also returned.

2 Specification

#include <nag.h>

#include <nagf01.h>

void	nag_matop_complex_gen_matrix_cond_sqrt (Integer n, Complex a[], Integer pda, double alpha, double condsa, NagError *fail)

3 Description

For a matrix with no eigenvalues on the closed negative real line, the principal matrix square root,

A^{1 / 2}

, of

A

is the unique square root with eigenvalues in the right half-plane.

The Fréchet derivative of a matrix function

A^{1 / 2}

in the direction of the matrix

E

is the linear function mapping

E

L (A, E)

such that

{(A + E)}^{1 / 2} - A^{1 / 2} - L (A, E) = o (‖A‖) .

The absolute condition number is given by the norm of the Fréchet derivative which is defined by

‖L (A)‖ := \max_{E \neq 0} \frac{‖L (A, E)‖}{‖E‖} .

The Fréchet derivative is linear in

E

and can therefore be written as

vec (L (A, E)) = K (A) vec (E),

where the

vec

operator stacks the columns of a matrix into one vector, so that

K (A)

n^{2} \times n^{2}

nag_matop_complex_gen_matrix_cond_sqrt (f01kdc) uses Algorithm 3.20 from Higham (2008) to compute an estimate

γ

such that

γ \leq {‖K (X)‖}_{F}

. The quantity of

γ

provides a good approximation to

{‖L (A)‖}_{F}

. The relative condition number,

κ_{A^{1 / 2}}

, is then computed via

κ_{A^{1 / 2}} = \frac{{‖L (A)‖}_{F} {‖A‖}_{F}}{{‖A^{1 / 2}‖}_{F}} .

κ_{A^{1 / 2}}

is returned in the argument condsa.

A^{1 / 2}

is computed using the algorithm described in Higham (1987). This is a version of the algorithm of Björck and Hammarling (1983). In addition, a blocking scheme described in Deadman et al. (2013) is used.

The computed quantity

α

is a measure of the stability of the relative residual (see Section 7). It is computed via

α = \frac{{‖A^{1 / 2}‖}_{F}^{2}}{{‖A‖}_{F}} .

4 References

Björck Å and Hammarling S (1983) A Schur method for the square root of a matrix Linear Algebra Appl. 52/53 127–140

Deadman E, Higham N J and Ralha R (2013) Blocked Schur Algorithms for Computing the Matrix Square Root Applied Parallel and Scientific Computing: 11th International Conference, (PARA 2012, Helsinki, Finland) P. Manninen and P. Öster, Eds Lecture Notes in Computer Science 7782 171–181 Springer–Verlag

Higham N J (1987) Computing real square roots of a real matrix Linear Algebra Appl. 88/89 405–430

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

5 Arguments

1: n – IntegerInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $n \geq 0$ .
2: a[ $\dim$ ] – ComplexInput/Output: Note: the dimension, dim, of the array a must be at least $pda \times n$ .

The $(i, j)$ th element of the matrix $A$ is stored in $a [(j - 1) \times pda + i - 1]$ .
On entry: the $n$ by $n$ matrix $A$ .

On exit: the $n$ by $n$ principal matrix square root $A^{1 / 2}$ . Alternatively, if $fail . code =$ NE_EIGENVALUES, contains an $n$ by $n$ non-principal square root of $A$ .
3: pda – IntegerInput: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
4: alpha – double *Output: On exit: an estimate of the stability of the relative residual for the computed principal (if $fail . code =$ NE_NOERROR) or non-principal (if $fail . code =$ NE_EIGENVALUES) matrix square root, $α$ .
5: condsa – double *Output: On exit: an estimate of the relative condition number, in the Frobenius norm, of the principal (if $fail . code =$ NE_NOERROR) or non-principal (if $fail . code =$ NE_EIGENVALUES) matrix square root at $A$ , $κ_{A^{1 / 2}}$ .
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALG_FAIL: An error occurred when computing the condition number. The matrix square root was still returned but you should use nag_matop_complex_gen_matrix_sqrt (f01fnc) to check if it is the principal matrix square root.
An error occurred when computing the matrix square root. Consequently, alpha and condsa could not be computed. It is likely that the function was called incorrectly.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_EIGENVALUES: $A$ has a negative or semisimple vanishing eigenvalue. A non-principal square root was returned.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pda \geq n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR: $A$ has a defective vanishing eigenvalue. The square root and condition number cannot be found in this case.

7 Accuracy

If the computed square root is

\tilde{X}

, then the relative residual

\frac{{‖A - {\tilde{X}}^{2}‖}_{F}}{{‖A‖}_{F}},

is bounded approximately by

n α ε

, where

ε

is machine precision. The relative error in

\tilde{X}

is bounded approximately by

n α κ_{A^{1 / 2}} ε

8 Parallelism and Performance

nag_matop_complex_gen_matrix_cond_sqrt (f01kdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_complex_gen_matrix_cond_sqrt (f01kdc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Approximately

3 \times n^{2}

of complex allocatable memory is required by the function.

The cost of computing the matrix square root is

85 n^{3} / 3

floating-point operations. The cost of computing the condition number depends on how fast the algorithm converges. It typically takes over twice as long as computing the matrix square root.

If condition estimates are not required then it is more efficient to use nag_matop_complex_gen_matrix_sqrt (f01fnc) to obtain the matrix square root alone. Condition estimates for the square root of a real matrix can be obtained via nag_matop_real_gen_matrix_cond_sqrt (f01jdc).

10 Example

This example estimates the matrix square root and condition number of the matrix

A = (\begin{array}{r} 29 + 35 i & 31 + 61 i & - 38 + 49 i & - 17 - 6 i \\ 52 - 59 i & 58 - 29 i & 97 + 39 i & - 32 + 15 i \\ 20 - 31 i & 44 - i & 37 + 19 i & - 26 + 19 i \\ - 70 + 72 i & - 90 + 8 i & - 87 - 43 i & 47 - 5 i \end{array}) .

NAG Library Function Documentnag_matop_complex_gen_matrix_cond_sqrt (f01kdc)